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Let $\epsilon_i$ be i.i.d. Rademacher random variables (i.e., $\epsilon_i$ takes value $\pm 1$ with equal probability). The upper bound $\mathbb{E} |\sum_{i=1}^n \epsilon_i| \le \sqrt{n}$ follows from Jensen's inequality. [Can this be sharpened easily?]

My main question: is there a simple lower bound for this expectation?

angryavian
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  • One knows that $E|\epsilon_1+\cdots+\epsilon_n|/\sqrt{n}\to E|Z|$ where $Z$ is standard normal hence $E|Z|=\sqrt{2/\pi}$. – Did Apr 13 '16 at 22:15
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    My answer here http://math.stackexchange.com/questions/883982/non-probabilistic-argument-for-divergence-of-the-simple-random-walk/884211#884211 gives an explicit formula for $\mathbb{E}\left|\sum_{i=1}^{2n} \epsilon_i\right|$. You can get upper and lower bounds if you follow the links there. –  Apr 13 '16 at 22:20
  • @Did Thanks, that is a good sanity check that I forgot. Do you know of any non-asymptotic bounds? I am hoping for a lower bound of the form $\ge c \sqrt{n}$. – angryavian Apr 13 '16 at 22:23
  • ?? Did you check @Byron's link? – Did Apr 13 '16 at 22:24
  • @ByronSchmuland Ah thanks! So for even $n$, we have $\ge \sqrt{n/2}$? – angryavian Apr 13 '16 at 22:25
  • @angryavian Yes, that is what I get. I moved my comment to an answer. –  Apr 13 '16 at 22:26
  • See https://www.wikiwand.com/en/Khintchine_inequality – durdi Oct 01 '20 at 14:43

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My answer here gives an explicit formula for $\mathbb{E}\left|\sum_{i=1}^{2n} \epsilon_i\right|$. You can get the lower bound $$\mathbb{E}\left|\sum_{i=1}^{2n} \epsilon_i\right|\geq \sqrt{n}, \tag1$$ if you combine (1) with the bound at the link there.

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